(1) Field of the Invention
This invention provides a more accurate and flexible method for producing spatial layouts of objects (including equipments, personnel, or combinations of equipments and personnel) under circumstances in which it is important to consider discrete spatial density in any rectangular or square lattice containing 100 or less density points. Thus, the relationship among objects in a particular space can be accurately determined to minimize crowding.
(2) Description of the Prior Art
The conventional formula to measure or model two-dimensional discrete spatial density; i.e., population density or physical crowding is defined as the average number of objects (n) per unit area of space (A): EQU D=n/A (1)
This definition has severe shortcomings since actual spatial orientation within a specified area is disregarded. As an example of this shortcoming, refer to FIG. 1 which displays three different configurations of objects or density points. In each case, the "perceived density" of the four points is obviously different. Since the number of points and area are identical in each depiction, there is a constant value of 0.25 for Population density. FIG. 2 depicts geometrically the population demographer's model of population density shown for the distributions in FIG. 1. FIG. 2 shows that each point occupies four space units (such as feet); hence, population density or physical crowdedness (D=n/A) equals one object per four square feet. FIG. 2 represents the model for each depiction of FIG. 1. However, large differences in perceived physical crowding clearly exist among the three configurations shown in FIG. 1.
A formula was then derived by the inventor to capture the differences shown in FIG. 1 more accurately by taking the actual spatial orientation of objects into account. See O'Brien, F., "A Crowding Index for Finite Populations", Perceptual and Motor Skills, February 1990, 70, pp. 3-11 which is incorporated into this disclosure in its entirety by reference.
This formula, referred to as the Population Density Index (PDI), is as follows: ##EQU1## where n=number of objects
A=the geometric area, and PA1 d=average Euclidean distance among all possible pairs of n objects. PA1 n=the number of objects within one area.
Basically, the above proposed formula is a generalization of the bivariate Euclidean distance formula. The derivation of the proposed formula is patterned on the well known square-root law used in the physical sciences.
Assume two objects are plotted on an X, Y Cartesian coordinate system with a fixed origin O. The mathematical distance between the two objects is measurable by simple analytic geometry using the Pythagorean distance formula: EQU d.sub.12 =[(X.sub.1 -X.sub.2).sup.2 +(Y.sub.1 -Y.sub.2).sup.2 ].sup.1/2( 3)
where (X.sub.1,Y.sub.1); (X.sub.2,Y.sub.2) represent each object's coordinates.
If, now, we conceive of n objects, each given coordinates within the same geometric plane such as a room, it is possible to generalize the above formula to obtain an average Euclidean distance among the n objects. The average Euclidean distance of n points, considered pairwise, is given by: ##EQU2## where d.sub.ij is the Euclidean distance between any two objects. Note that for n=2 objects, d;d.sub.12 are equivalent.
The last step in deriving a density index is to scale d to adjust for a given number of objects residing within a specific area. A proposed general formula based on the square-root inverse law for distances incorporating size of area and the number of objects is: ##EQU3## where A=the geometric area in which objects reside, and
Dimensional analysis, as well as empirical Monte Carlo simulation investigations, of .DELTA. shows that the units are: ##EQU4## Essentially, .DELTA. is the average pairwise Euclidean distance among n objects scaled for a given unit area. As will become evident in the following numerical example, .DELTA. is inversely related to the average geometric distances among n points. Calculating the reciprocal of .DELTA., 1/.DELTA. will make the relationship monotonically increasing, that is, the more densely packed the objects, the higher the value of the index. This reciprocal of .DELTA., or 1/.DELTA., is arbitrarily referred to as the population density index, or PDI. The units for PDI are .sqroot.n/A.
A computational example is provided with the aid of FIG. 3. For four points, there are 4.times.3/2=6 pairwise distances to calculate. The coordinate points for the 4 units are (1,1), (1,3), (2,4) and (3,2). The area shown is 16 units. Applying .DELTA., ##EQU5## Calculating the reciprocal of .DELTA. and multiplying by 10 to give integer results, PDI=2.3.
The .DELTA. index appears to be valid even when areas differ by a large amount. To demonstrate this, consider FIGS. 4A and 4B. The average Euclidean distances are identical (1.6) in each situation depicted. The smaller value of .DELTA. in FIG. 4A (3.7) is in accord with the basic interpretation of .DELTA., that is, the smaller the value of .DELTA., the more densely packed are the points relative to the allowed area. The results also correspond to the intuitive notion of density.
The proposed crowding index, .DELTA. or PDI, should be interpreted as a relative measure much like a standard deviation in statistics. The theoretical mathematical minimum value of .DELTA. or PDI is always 0, a condition realizable with dimensionless points but not realizable with solid objects such as people.
The maximum value depends on the number of objects and the geometric area. Beyond three or four objects, it becomes difficult and perhaps meaningless to attempt calculating a precise maximum value of .DELTA. or PDI. For these reasons, hypothetical minimum and maximum bounds of the PDI formula are derived below and presented as an integral component of this disclosure. Three additional properties derived from the square-root law for average distances of .DELTA. or PDI appear to be critical to the usefulness and interpretability of the index: 1) for constant area, PDI varies directly with the number of objects; 2) for a constant number of objects, PDI varies indirectly with area; and 3) for a constant number of objects and constant area, PDI varies indirectly with distance. Small sample Monte Carlo simulations performed by the inventor have supported these square-root properties for the PDI formula. The values of PDI computed from randomly selected uniform distributions correlated 0.96 with the conventional formula for population density (n/A).
In addition, the PDI formula can be evaluated on three key scientific criteria. First, the model is very simple. It connects population density to three key variables--distance, number of points and area--through an equation that can be readily calculated. Second, the formula is justified by mathematical analysis. The inverse square-root properties of the index stated as conjectures are very reasonable and provide a context for prediction and explanation of observed results. Monte Carlo simulations support each conjecture, thereby providing preliminary justification until large scale simulations can be conducted. Third, the formula has been tested and verified by empirical research. The use of the formula in hypothetical military settings has produced results that were readily interpretable and which correlated with qualitative estimates of crowding made by independent expert observers.
The population density formula attempts to express differences such as those shown in FIG. 1 more accurately than the conventional population density formula. Since the index can vary widely, as indicated in FIG. 1, it was necessary to develop a new model to predict minimum and maximum bounds of the population density index values. It was in this light that the present invention was conceived and has now been reduced to practice.